2 Definition of admissible sets and argumentation stages

In this section, we give the main definitions of the admissible set approach, taken from Dung (1995), and the argumentation stage approach, adapted from Verheij (1995a, 1995b).

Argumentation depends on the arguments that can be taken into account, and on which arguments challenge other arguments. In this paper, arguments are considered as abstract unstructured objects. Defeaters represent which arguments challenge other arguments. This leads to the following definition of an argumentation theory.[2]

Definition 1.

An argumentation theory is a pair (Arguments, Defeaters), where Arguments is any set, and Defeaters is a subset of Arguments ( Arguments. The elements of Arguments are the arguments of the theory, the elements of Defeaters the defeaters. In a defeater (Arg, Arg'), the argument Arg is the challenging argument, and Arg' the challenged argument.

The following definitions and results depend on a not explicitly mentioned argumentation theory (Arguments, Defeaters), unless specified otherwise.

Definition 2 summarizes some of Dung's definitions. For an extended discussion, we refer to the original paper (Dung, 1995). Central in his definitions is the notion of an acceptable argument. An argument Arg is acceptable with respect to some set of arguments Args if all arguments that challenge the argument Arg are themselves challenged by an argument in the set Args.

Definition 2. (Dung, 1995)

(1) A set of arguments Args is conflict-free if there is no defeater (Arg, Arg'), such that Arg and Arg' both are elements of Args.

(2) An argument Arg is acceacceptable with respect to a set of arguments Args if for all arguments Arg' of the theory the following holds:

If (Arg', Arg) is a defeater, then there is an argument Arg'' in Args, such that (Arg'', Arg') is a defeater.

(3) A set of arguments Args is admissible if it is conflict-free and all arguments in Args are acceptable with respect to Args.

(4) A preferred extension of an argumentation theory is an admissible set of arguments, that is maximal with respect to set inclusion.

(5) A conflict-free set of arguments Args is a stable extension of an argumentation theory if for any argument of the theory Arg that is not in Args, there is an argument Arg' in Args, such that (Arg', Arg) is a defeater.[3]

An argumentation theory has at least one preferred extension, since the empty set {} is admissible, and unions of increasing sequences of admissible sets are admissible.[4] A theory does not always have a stable extension. For instance, the argumentation theory ({alfa}, {(alfa, alfa)}) has the empty set {} as unique preferred extension, which is not stable. A theory can have more than one preferred extension. For instance, the argumentation theory ({alfa, beta}, {(alfa, beta), (beta, alfa)}) has the preferred (and stable) extensions {alfa} and {beta} (see section 4.2).

In the following definition, the argumentation stage approach is summarized. It is a restricted version of the definitions by Verheij (1995a, 1995b).[5] The adaptation was made to make the relations with Dung's admissible set approach clearly visible.

Intuitively, an argumentation stage is characterized by the arguments that have been taken into account, and by the statuses of these arguments. Each argument has one of two statuses: either undefeated or defeated. Formally, an argumentation stage is a status assignment, that satisfies a constraint: Any argument challenged by an undefeated argument must be defeated, and any defeated argument must be challenged by an undefeated argument. A stage extension is now an argumentation stage in which a maximal number of arguments is taken into account, i.e., a stage that has maximal range.

Definition 3.

(1) A defeat status assignment is a pair of disjoint sets of arguments. In a defeat status assignment (UndefeatedArgs, DefeatedArgs), the arguments in UndefeatedArgs are undefeated, those in DefeatedArgs are defeated. The union of the sets UndefeatedArgs and DefeatedArgs is the range of the defeat status assignment.

(2) An argumentation stage (or stage, for short) is a defeat status assignment (UndefeatedArgs, DefeatedArgs), such that for each argument Arg in its range the following holds:

Arg is an element of DefeatedArgs if and only if there is an argument Arg' in UndefeatedArgs, such that (Arg', Arg) is a defeater.

(3) A stage extension is an argumentation stage (UndefeatedArgs, DefeatedArgs), such that there is no argumentation stage with larger range. A stage extension is complete if all arguments of the argumentation theory are in its range.[6]

An argumentation theory does not always have a stage extension. For instance, the argumentation theory ({alfai | i = 0, 1, 2, ...}, {(alfai, alfaj) | i > j}) has no stage extension. It has several sensible stages, though, such as ({alfai}, {alfaj | i > j}) for any i = 0, 1, 2, ..., while its preferred extension {} is its only admissible set of arguments (see section 4.5). A theory can have more than one stage extension. For instance, the argumentation theory ({alfa, beta}, {(alfa, beta), (beta, alfa)}) has the (complete) stage extensions ({alfa}, {beta}) and ({beta}, {alfa}).

Each argument in an admissible set must be defended against all challenging arguments. In an argumentation stage, each undefeated argument must only be defended against the challenging arguments that have been taken into account, i.e., against the arguments in the range of the stage. This is intuitively the main difference between the two approaches.